Metamath Proof Explorer


Theorem frege79

Description: Distributed form of frege78 . Proposition 79 of Frege1879 p. 63. (Contributed by RP, 1-Jul-2020) (Revised by RP, 3-Jul-2020) (Proof modification is discouraged.)

Ref Expression
Hypotheses frege79.x
|- X e. U
frege79.y
|- Y e. V
frege79.r
|- R e. W
frege79.a
|- A e. B
Assertion frege79
|- ( ( R hereditary A -> A. a ( X R a -> a e. A ) ) -> ( R hereditary A -> ( X ( t+ ` R ) Y -> Y e. A ) ) )

Proof

Step Hyp Ref Expression
1 frege79.x
 |-  X e. U
2 frege79.y
 |-  Y e. V
3 frege79.r
 |-  R e. W
4 frege79.a
 |-  A e. B
5 1 2 3 4 frege78
 |-  ( R hereditary A -> ( A. a ( X R a -> a e. A ) -> ( X ( t+ ` R ) Y -> Y e. A ) ) )
6 ax-frege2
 |-  ( ( R hereditary A -> ( A. a ( X R a -> a e. A ) -> ( X ( t+ ` R ) Y -> Y e. A ) ) ) -> ( ( R hereditary A -> A. a ( X R a -> a e. A ) ) -> ( R hereditary A -> ( X ( t+ ` R ) Y -> Y e. A ) ) ) )
7 5 6 ax-mp
 |-  ( ( R hereditary A -> A. a ( X R a -> a e. A ) ) -> ( R hereditary A -> ( X ( t+ ` R ) Y -> Y e. A ) ) )